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| Mirrors > Home > MPE Home > Th. List > Mathboxes > grlimprop2 | Structured version Visualization version GIF version | ||
| Description: Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025.) |
| Ref | Expression |
|---|---|
| grlimprop.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| grlimprop.w | ⊢ 𝑊 = (Vtx‘𝐻) |
| grlimprop2.n | ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣) |
| grlimprop2.m | ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣)) |
| grlimprop2.i | ⊢ 𝐼 = (iEdg‘𝐺) |
| grlimprop2.j | ⊢ 𝐽 = (iEdg‘𝐻) |
| grlimprop2.k | ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} |
| grlimprop2.l | ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} |
| Ref | Expression |
|---|---|
| grlimprop2 | ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | grlimdmrel 47947 | . . . . 5 ⊢ Rel dom GraphLocIso | |
| 2 | 1 | ovrcl 7472 | . . . 4 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V)) |
| 3 | id 22 | . . . 4 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) | |
| 4 | df-3an 1089 | . . . 4 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) ↔ ((𝐺 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻))) | |
| 5 | 2, 3, 4 | sylanbrc 583 | . . 3 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻))) |
| 6 | grlimprop.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 7 | grlimprop.w | . . . 4 ⊢ 𝑊 = (Vtx‘𝐻) | |
| 8 | grlimprop2.n | . . . 4 ⊢ 𝑁 = (𝐺 ClNeighbVtx 𝑣) | |
| 9 | grlimprop2.m | . . . 4 ⊢ 𝑀 = (𝐻 ClNeighbVtx (𝐹‘𝑣)) | |
| 10 | grlimprop2.i | . . . 4 ⊢ 𝐼 = (iEdg‘𝐺) | |
| 11 | grlimprop2.j | . . . 4 ⊢ 𝐽 = (iEdg‘𝐻) | |
| 12 | grlimprop2.k | . . . 4 ⊢ 𝐾 = {𝑥 ∈ dom 𝐼 ∣ (𝐼‘𝑥) ⊆ 𝑁} | |
| 13 | grlimprop2.l | . . . 4 ⊢ 𝐿 = {𝑥 ∈ dom 𝐽 ∣ (𝐽‘𝑥) ⊆ 𝑀} | |
| 14 | 6, 7, 8, 9, 10, 11, 12, 13 | isgrlim2 47950 | . . 3 ⊢ ((𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ (𝐺 GraphLocIso 𝐻)) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) |
| 15 | 5, 14 | syl 17 | . 2 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹 ∈ (𝐺 GraphLocIso 𝐻) ↔ (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖))))))) |
| 16 | 15 | ibi 267 | 1 ⊢ (𝐹 ∈ (𝐺 GraphLocIso 𝐻) → (𝐹:𝑉–1-1-onto→𝑊 ∧ ∀𝑣 ∈ 𝑉 ∃𝑓(𝑓:𝑁–1-1-onto→𝑀 ∧ ∃𝑔(𝑔:𝐾–1-1-onto→𝐿 ∧ ∀𝑖 ∈ 𝐾 (𝑓 “ (𝐼‘𝑖)) = (𝐽‘(𝑔‘𝑖)))))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∀wral 3061 {crab 3436 Vcvv 3480 ⊆ wss 3951 dom cdm 5685 “ cima 5688 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 Vtxcvtx 29013 iEdgciedg 29014 ClNeighbVtx cclnbgr 47805 GraphLocIso cgrlim 47943 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-1o 8506 df-map 8868 df-vtx 29015 df-iedg 29016 df-clnbgr 47806 df-isubgr 47847 df-grim 47864 df-gric 47867 df-grlim 47945 |
| This theorem is referenced by: (None) |
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